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Existence and sharp asymptotic behavior of positive decreasing solutions of a class [4pt] of differential systems with power-type nonlinearities

Jaroslav JarošKusano Takaŝi — 2014

Archivum Mathematicum

The system of nonlinear differential equations x ' + p 1 ( t ) x α 1 + q 1 ( t ) y β 1 = 0 , y ' + p 2 ( t ) x α 2 + q 2 ( t ) y β 2 = 0 , A is under consideration, where α i and β i are positive constants and p i ( t ) and q i ( t ) are positive continuous functions on [ a , ) . There are three types of different asymptotic behavior at infinity of positive solutions ( x ( t ) , y ( t ) ) of (). The aim of this paper is to establish criteria for the existence of solutions of these three types by means of fixed point techniques. Special emphasis is placed on those solutions with both components decreasing to zero as t , which can be...

Asymptotic analysis of positive solutions of generalized Emden-Fowler differential equations in the framework of regular variation

Jaroslav JarošKusano TakaŝiJelena Manojlović — 2013

Open Mathematics

Positive solutions of the nonlinear second-order differential equation ( p ( t ) | x ' | α - 1 x ' ) ' + q ( t ) | x | β - 1 x = 0 , α > β > 0 , are studied under the assumption that p, q are generalized regularly varying functions. An application of the theory of regular variation gives the possibility of obtaining necessary and sufficient conditions for existence of three possible types of intermediate solutions, together with the precise information about asymptotic behavior at infinity of all solutions belonging to each type of solution classes.

Asymptotic properties of solutions of second order quasilinear functional differential equations of neutral type

Takaŝi KusanoPavol Marušiak — 2000

Mathematica Bohemica

This paper establishes existence of nonoscillatory solutions with specific asymptotic behaviors of second order quasilinear functional differential equations of neutral type. Then sufficient, sufficient and necessary conditions are proved under which every solution of the equation is either oscillatory or tends to zero as t .

Picone's Identity for Ordinary Differential Operators of Even Order

Takasi KusanoNorio Yoshida — 1975

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti

In questo lavoro la ben nota identità di M. Picone è generalizzata agli operatori differenziali ordinari autoaggiunti di ordine superiore. Tale identità generalizzata è impiegata per conseguire teoremi di confronto del tipo di Sturm e criteri di non oscillazione per le soluzioni di equazioni (o diseguaglianze) relative a tali operatori.

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